Two meson systems with Ginsparg-Wilson valence quarks

Unphysical effects associated with finite lattice spacing and partial quenching may lead to the presence of unphysical terms in chiral extrapolation formulas. These unphysical terms must then be removed during data analysis before physical predictions can be made. In this work, we show that through next-to-leading order, there are no unphysical counterterms in the extrapolation formulas, expressed in lattice-physical parameters, for meson scattering lengths in theories with Ginsparg-Wilson valence quarks. Our work applies to most sea quark discretizations, provided that chiral perturbation theory is a valid approximation. We demonstrate our results with explicit computations and show that, in favorable circumstances, the extrapolation formulas do not depend on the unknown constant $C_{\mathrm{Mix}}$ appearing at lowest order in the mixed action chiral Lagrangian. We show that the $I=1$ $KK$ scattering length does not depend on $C_{\mathrm{Mix}}$ in contrast to the $I=3/2$ $K\pi$ scattering length. In addition, we show that these observables combined with $f_K/f_{\pi }$ and the $I=2$ $\pi \pi$ scattering length share only two linearly independent sets of counterterms, providing a means to test the mixed action theory at one lattice spacing. We therefore make a prediction for the $I=1$ $KK$ scattering length.